I'm not sure what a dimmable brake light is but I can supply some information about LED brake lights.
Filament bulbs, when they get hotter, increase their resistance and this reduces the increase in current that comes from an increase in voltage. This is quite useful when the 12V supply of a car can vary from <12V to >15V.
LEDs in contrast have an exponential increase in current with increasing voltage once the threshold voltage is exceeded. Because of this most LED brakelights include a constant current regulator to keep the LED at its maximum safe current no matter how the 12V supply varies.
If you have a variable dimmable voltage supply fighting with a regulated constant current supply there is a good chance that the constant current supply will lose.
LED lights designed to dim with a variable voltage will either have very complicated electronics or very crude and inefficient resistor regulation where ~90% of the power to the LED is dissipated in a resistor.
But I suspect you may just have the wrong pins for the socket (as well as a dimmable problem).
Good Luck. Colin